A Central Limit Theorem for Stochastic Saddle Point Optimization

In this work, we study the Uncertainty Quantification (UQ) of an algorithmic estimator of the saddle point of a strongly-convex strongly-concave objective. Specifically, we show that the averaged iterates of a Stochastic Extra-Gradient (SEG) method for a Saddle Point Problem (SPP) converges almost surely to the saddle point and follows a Central Limit Theorem (CLT) with optimal covariance under two different noise settings, namely the martingale-difference noise and the state-dependent Markov noise. To ensure the stability of the algorithm dynamics under the state-dependent Markov noise, we propose a variant of SEG with truncated varying sets. Our work opens the door for online inference of SPP with numerous potential applications in GAN, robust optimization, and reinforcement learning to name a few. We illustrate our results through numerical experiments.